Chapter 13 Game Theory

Topics Concepts and Definitions Static Games Solution concepts: Dominance, Nash Dynamic Games. Auctions (optional).

Game Theory Game theory - a set of tools that economists, political scientists, military analysts and others use to analyze decision making by players who interact strategically

Application Examples Matching of residents and hospitals, kidney donors and recipients… (latest Nobel in economics) Auction design for broadband licenses Military: mutually assured destruction (MAD) And many more!

Example Jules and Vincent are mobsters; they have killed someone, are being arrested, but evidence only enough for illegal weapon holding Police sit them in separate rooms, offering each to get jail reduction if they implicate the other 2 years for weapons; 15 years for murder; 1 year off for both confessing, 0 if you are the only one to implicate the other

An Overview of Game Theory Game - any interaction between players (firms) in which strategic behavior plays a major role. Action - a move that a player makes at a specified stage of a game, such as how much output a firm produces in the current period.

An Overview of Game Theory (cont.) Strategy - a battle plan that specifies the action that a player will make conditional on the information available at each move and for any possible contingency. Payoffs - players’ valuations of the outcome of the game, such as profits for firms or utilities for individuals.

Main Assumptions  players are interested in maximizing their payoffs.  all players have common knowledge:  about the rules of the game,  that each player’s payoff depends on actions taken by all players,  that all players want to maximize payoffs,  that all players know that all players know the payoffs and  that their opponents know they are payoff maximizing, and so on.

An Overview of Game Theory (cont.) Strategic behavior - a set of actions a firm takes to increase profit, taking into account the possible actions of other firms Strategic interdependence - a player’s optimal strategy depends on the actions of others.

An Overview of Game Theory (cont.) Rules of the game - regulations that determine the timing of players’ moves and the actions that players can make at each move

Information Definitions Complete information - the situation where the payoff function is common knowledge among all players Perfect information - the situation where the player who is about to move knows the full history of the play of the game to this point, and that information is updated with each subsequent action

Static vs. Dynamic Games Static game - game in which each player acts only once and the players act simultaneously (or, at least, each player acts without knowing rivals’ actions). Dynamic game - game in which players move either sequentially or repeatedly

Normal-Form Games Normal form - representation of a static game with complete information specifies the players in the game, their possible strategies, and the payoff function that identifies the players’ payoffs for each combination of strategies

Normal-Form Games: Example Two airlines: American and United. Each airline can take only one of two possible actions:  Each can fly either 64 or 48 thousand passengers between Chicago and Los Angeles per quarter.

Normal-Form Games: Example (cont.) Because the firms choose their strategies simultaneously, each firm selects a strategy that maximizes its profit given what it believes the other firm will do. The firms are playing a noncooperative game of imperfect information.

Table 13.1 Profit Matrix for a Quantity- Setting Game: Dominant Strategy

Predicting a Game’s Outcome Dominant strategy - a strategy produces a higher payoff than any other strategy the player can use for every possible combination of its rivals’ strategies.

Predicting a Game’s Outcome If United chooses the high-output strategy (q U = 64), American’s high-output strategy maximizes its profit. If United chooses the low-output strategy (q U = 48), American’s high-output strategy maximizes its profit. Thus, the high-output strategy is American’s dominant strategy.  Whichever strategy United uses, American’s profit is higher if it uses its high-output strategy.

Predicting a Game’s Outcome A striking feature of this game is that the players choose strategies that do not maximize their joint profit. Prisoners’ dilemma - a game in which all players have dominant strategies that result in profits (or other payoffs) that are inferior to what they could achieve if they used cooperative strategies

Iterated Dominance Dominated strategy – a strategy that produces a lower payoff for every possible combination of its rivals’ strategies than some other strategy the player can use Iterated Dominance – Eliminate dominated strategies one at a time

Table 13.2 Profit Matrix for a Quantity- Setting Game: Iterated Dominance

Best Response Best response - the strategy that maximizes a player’s payoff given his beliefs about his rivals’ strategies.

Nash Equilibrium Nash equilibrium - a set of strategies such that, when all other players use these strategies, no player can obtain a higher payoff by choosing a different strategy – no incentive to deviate All players use best-response strategies to others’ best response strategies

Nash Equilibrium Always exists Only solution concept where beliefs are consistent with actual strategies

Best Responses and Nash Equilibrium In a game without dominant strategies, calculate best responses to determine Nash equilibrium

Failure to Maximize Joint Payoffs In the Nash equilibrium of the first advertising game, firms maximize joint profits. In the second, they do not

Multiple Nash Equilibria, No Nash Equilibrium, and Mixed Strategies Pure strategy - each player chooses an action with certainty  assigns a probability of 1 to a single action. Mixed strategy - a firm (player) chooses among possible actions according to probabilities it assigns.  probability distribution over actions

Multiple Nash Equilibria, No Nash Equilibrium, and Mixed Strategies (cont.) Suppose that two firms are considering opening gas stations at a highway rest stop that has no gas stations. There’s enough physical space for at most two gas stations.

Table 13.4 Simultaneous Entry Game

Multiple Nash Equilibria, No Nash Equilibrium, and Mixed Strategies (cont.) This game has two Nash equilibria in pure strategies:  Firm 1 enters and Firm 2 does not enter, or  Firm 2 enters and Firm 1 does not enter. How do the players know which (if any) Nash equilibrium will result?  They don’t know.

Mixed-strategy Nash Equilibrium Every finite game has at least one Nash Equilibrium – possibly in mixed strategies Assign a probability to each action Expected payoff for opponent has to be equal for all actions in its mixed strategy If true for both, then Nash Equilibrium

Example: Gas Station Entry Use probability θ for not entering, (1 - θ) for entering Opponent’s expected payoff is then  Not enter: θ*0 + (1 - θ)*0 = 0  Enter:θ*1 + (1 - θ)*(-1) = 2*θ - 1  Setting them equal: 0 = 2*θ - 1→ θ = 1/2 Game is symmetric so if each player uses the strategy {Pr(e)=1/2, Pr(n)=1/2}, that is a Nash Equilibrium in mixed strategies

Table 13.4 Simultaneous Entry Game

Solved Problem 13.2 Mimi wants to support her son Jeff if he looks for work but not otherwise. Jeff wants to try to find a job only if Mimi will not support his life of indolence. If they choose actions simultaneously, what are the pure- or mixed-strategy equilibria?

Solved Problem 13.2

Example 2: Shirking Jeff: probability θ J for working, (1 – θ J ) for loafing Mimi: θ M for support, (1 – θ M ) not support Mimi indifferent if  θ J *4 + (1 – θ J )*(-1) = θ J *(-1) + (1 – θ J )*0  θ J = 1/6 Jeff indifferent if  θ M *2 + (1 – θ M )*1 = θ M *4 + (1 – θ M )*0  θ M = 1/3 Nash Equilibrium in mixed strategies for {Pr(w)=1/6,Pr(l)=5/6} and {Pr(s)=1/3,Pr(n)=2/3}

Dynamic Games Dynamic games - players move sequentially or move simultaneously repeatedly over time Extensive form - specifies the n players, the sequence in which they make their moves, the actions they can take at each move, the information that each player has about players’ previous moves, and the payoff function over all the possible strategies

Subgames Subgame – all subsequent decisions that players may make given the actions already taken and corresponding payoffs  One subgame for every node of information  (Game itself is also a subgame of itself)

Sequential Game Two-stage game:  In the first stage, Player 1 moves.  In the second stage, Player 2 observes Player 1’s actions and then makes a choice based on this observed action.  Game ends with the players’ receiving payoffs based on their actions.

Sequential Game Same airline game as before.  Assume that American and United Airlines can choose output levels of 96, 64, and 48 thousand passengers per quarter.  American moves first and United can observe this choice before making its own choice

Figure 13.1 Airlines Game Tree

Compare to Table 13.2, Profit Matrix for a Quantity-Setting Game

Sequential Game (cont.) Subgame perfect Nash equilibrium - players’ strategies are a Nash equilibrium in every subgame. Backward induction - first determine the best response by the last player to move, next determine the best response for the player who made the next to-last move, then repeat the process back to the move at the beginning of the game

Sequential Game (cont.) How should American, the leader, select its output in the first stage?  For each possible quantity it can produce, American predicts what United will do and picks the output level that maximizes its own profit.

Sequential Game (cont.) The subgame perfect Nash equilibrium requires players to believe that their opponents will act optimally—in their own best interests. Not all Nash equilibria are subgame perfect Nash equilibria. We focus on the latter for dynamic games

Credibility Why doesn’t American announce that it will produce output q A =96 to induce United to produce output q U =48?  when the firms move simultaneously, United doesn’t believe American’s warning that it will produce a large quantity, because it is not in American’s best interest to produce that large a quantity of output.

Credibility (cont.) Credible threat - an announcement that a firm will use a strategy harmful to its rival and that the rival believes because the firm’s strategy is rational in the sense that it is in the firm’s best interest to use it

Credibility (cont.) Why commitment makes a threat credible?  “burning bridges.” - If the general burns the bridge behind the army so that the troops can only advance and not retreat, the army becomes a more fearsome foe

Repeated Game Static games that are repeated - in each period, there is a single stage:  Both players move simultaneously. Player 1’s move in period t precedes Player 2’s move in period t + 1; hence, the earlier action may affect the later one.  The players know all the moves from previous periods, but they do not know each other’s moves within any one period because they all move simultaneously.

Repeated Game (cont.) Suppose now that the airlines’ single- period prisoners’ dilemma game is repeated quarter after quarter.  If they play a single-period game, each firm takes its rival’s strategy as a given and assumes that it cannot affect that strategy. In a repeated game, a firm can influence its rival’s behavior by signaling and threatening to punish.

Auctions Auction - a sale in which property or a service is sold to the highest bidder. Three key components:  the number of units being sold,  the format of the bidding, and  the value that potential bidders place on the good.

Format of Bidding English auction - The auctioneer starts the bidding at the lowest price that is acceptable to the seller and then repeatedly encourages potential buyers to bid more than the previous highest bidder.

Format of Bidding (cont.) Dutch auction - ends dramatically with the first “bid.”  The seller starts by asking if anyone wants to buy at a relatively high price.  The seller reduces the price by given increments until someone accepts the offered price and buys at that price.

Format of Bidding (cont.) Sealed-bid auction - everyone submits a bid simultaneously without seeing anyone else’s bid (for example, by submitting each bid in a sealed envelope), and the highest bidder wins.  first-price auction - the winner pays its own highest bid.  second-price auction - the winner pays the amount bid by the second-highest bidder.

Value Private value - If each potential bidder places a different personal value on the good. Common value - good that has the same fundamental value to everyone, but no buyer knows exactly what that common value is.

Bidding Strategies in Private-Value Auctions In a traditional sealed-bid, second-price auction, bidding your highest value weakly dominates all other bidding strategies:  The strategy of bidding your maximum value leaves you as well off as, or better off than, bidding any other value.

Bidding Strategies in Private-Value Auctions (cont.) Should you ever bid more than your value?  Suppose that you bid $120. Your value is $100. There are three possibilities. If the highest bid of your rivals is greater than $120, then you do not buy the good and receive no consumer surplus. If the highest alternative bid is less than $100, then you win and receive the same consumer surplus that you would have received had you bid $100. if the highest bid by a rival were an amount between $100 and $120—say, $110—then bidding more than your maximum value causes you to win, but you purchase the good for more than you value it, so you receive negative consumer surplus: −$10 (= $100 − $110).

Bidding Strategies in Private-Value Auctions (cont.) Should you ever bid less than your maximum value, say, $90?  No, because you only lower the odds of winning without affecting the price that you pay if you do win. Thus, you do as well or better by bidding your value than by over- or underbidding.

English Auction Strategy The seller uses an English auction to sell the carving to bidders with various private values.  Your best strategy is to raise the current highest bid as long as your bid is less than the value you place on the good, $100.

Winner’s Curse Winner’s curse - auction winner’s bid exceeds the common-value item’s value If bidders are right on average, then highest bid likely to be above true value Example: Jar full of coins Example: early auctions for drilling rights